Question

2.   The president’s support among Republicans is very high. However, a political scientist wants to know whether fewer people these days are identifying as Republicans. In early November 2016, 27% of adult Americans identified as Republican.
A polling company selects a random sample of 1830 adult Americans, and finds that 435 identify themselves as Republicans.
i.   Check the conditions. Be sure to explain your answers, when needed.
ii.   Write hypotheses (be sure to define p, like “p = the proportion of ______”).
iii.   With α = 0.01, what critical value is needed to reject H0?
iv.   Perform the calculations for the test (find test statistic z, find p-value).
v.   Write a sentence on whether the proportion of Americans identifying as Republican has significantly decreased (still using α = 0.01).

Answer 1

Ans:

i)

n=1830

p₀=0.27

np₀(1-p₀)=1830*0.27*(1-0.27)=360.69>10,so we can use z test.

ii)

Let p = the proportion of  adult Americans identified as Republican

$H_0:p=0.27$

$H_a:p<0.27$

iii)

alpha=0.01

critical z value=normsinv(0.01)=-2.33

iv)

sample proportion=435/1830=0.2377

Test statistic:

z=(0.2377-0.27)/sqrt(0.27*(1-0.27)/1830)

z=-3.11

p-value=P(z<-3.11)=0.0009

v)As,p-value<0.01,we reject the null hypothesis.

There is sufficient evidence to conclude that the proportion of Americans identifying as Republican has significantly decreased